We will need more information about your application. If you can share a block diagram and the control signals between FX3 and ASIC we can understand your application better.
04b3:00f3. is the VID/PID for bootloader. It may mean that the firmware is not programmed onto the FX3 yet. Please program the .img into the FX3 RAM and see if this VID/PID changes. You can also use non-volatile SPI Flash/I2C EEPROM to boot from, but you will need to change the PMODE pins accordingly for the FX3 to recognize the boot mode. This settings are mentioned in the FX3 datasheet (If you are using explorer kit, you need to connect the J4 for USB boot. Program the EEPROM, and the remove the J4 jumper to make it boot from on-board EEPROM).